Polynomial bound for the nilpotency index of finitely generated nil algebras

Domokos, Mátyás

doi:10.2140/ant.2018.12.1233

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Polynomial bound for the nilpotency index of finitely generated nil algebras

Mátyás Domokos

Vol. 12 (2018), No. 5, 1233–1242

DOI: 10.2140/ant.2018.12.1233

Abstract

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating system of the ring of simultaneous conjugation invariants of tuples of $n$ -by- $n$ matrices. This is deduced from a result of Zubkov. As a consequence, a recent degree bound due to Derksen and Makam for the generators of the ring of matrix invariants yields an upper bound for the nilpotency index of a finitely generated nil algebra that is polynomial in the number of generators and the nil index. Furthermore, a characteristic free treatment is given to Kuzmin’s lower bound for the nilpotency index.

Keywords

nil algebra, nilpotent algebra, matrix invariant, degree bound

Mathematical Subject Classification 2010

Primary: 16R10

Secondary: 13A50, 15A72, 16R30

Milestones

Received: 27 June 2017

Accepted: 29 March 2018

Published: 31 July 2018

Authors

Mátyás Domokos
MTA Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences Budapest Hungary

Vol. 12, No. 5, 2018